Materials Science and Engineering - Electronic and Mechanical Properties of Materials Part 2 pot

3.225 21 Example: Conductivity Engineering • Objective: increase strength of Cu but keep conductivity high τ τ µ τ σ v m e m ne = == l 2 Scattering length connects scattering time to microstructure Dislocation (edge) l decreases, τ decreases, σ decreases e- © E.A. Fitzgerald-1999 3.225 22 • Can increase strength with second phase particles • As long as distance between second phase< l, conductivity marginally effected Example: Conductivity Engineering L S L+S Sn Cu L X Cu α β α+L β+L α+β S microstructure Material not strengthened, conductivity decreases α β dislocation L L>l Dislocation motion inhibited by second phase; material strengthened; conductivity about the same © E.A. Fitzgerald-1999 11 3.225 23 • Scaling of Si CMOS includes conductivity engineering • One example: as devices shrink… – vertical field increases – τ decreases due to increased scattering at SiO 2 /Si interface – increased doping in channel need for electrostatic integrity: ionized impurity scattering – τ SiO2 <τ impurity if scaling continues ‘properly’ Example: Conductivity Engineering E vert Ionized impurities (dopants) S D G SiO 2 © E.A. Fitzgerald-1999 3.225 24 Determining n and µ: The Hall Effect V x , E x I, J x B z + + + + + + + + + + + BvqEqF r r rr ×+= zDy BevF −= E y yy eEF −= In steady state, HZDY EBvE == , the Hall Field Since v D =-J x /en, ZXHZxH BJRBJ ne E =−= 1 ne R H 1 −= µσ ne = © E.A. Fitzgerald-1999 12 3.225 25 Experimental Hall Results on Metals • Valence=1 metals look like free-electron Drude metals • Valence=2 and 3, magnitude and sign suggest problems © E.A. Fitzgerald-1999 13 3.225 1 Response of Free e- to AC Electric Fields • Microscopic picture e - ti OZ eEE ω− = B=0 in conductor, and )()( BFEF rrrr >> ti eeE tp dt tdp ω τ − −−= 0 )()( ti eptp ω− = 0 )( 0 0 0 eE p pi −−=− τ ω try τ ω 1 0 0 − = i eE p ω>>1/τ, p out of phase with E ω<<1/τ, p in phase with E ω i eE p 0 0 = 0, →∞→ pω τ 00 eEp = © E.A. Fitzgerald-1999 3.225 2 Complex Representation of Waves sin(kx-ωt), cos(kx-ωt), and e -i(kx-ωt) are all waves e -i(kx-ωt) is the complex one and is the most general real imaginary A Acosθ iAsinθ θ e iθ =cosθ+isinθ © E.A. Fitzgerald-1999 1 3.225 3 • Momentum represented in the complex plane Response of e- to AC Electric Fields real imaginary p p (ω<<1/τ) p ( ω >>1/ τ ) Instead of a complex momentum, we can go back to macroscopic and create a complex J and σ ωτ i eJtJ − = 0 )( 0 2 0 0 ) 1 ( E im ne m nep nevJ ω τ − = − =−= m ne i τ σ ωτ σ σ 2 0 0 , 1 = − = © E.A. Fitzgerald-1999 3.225 4 • Low frequency (ω<<1/τ) – electron has many collisions before direction change – Ohm’s Law: J follows E, σ real • High frequency (ω>>1/τ) – electron has nearly 1 collision or less when direction is changed – J imaginary and 90 degrees out of phase with E, σ is imaginary Response of e- to AC Electric Fields Qualitatively: ωτ<<1, electrons in phase, re-irradiate, E i =E r +E t , reflection ωτ>>1, electrons out of phase, electrons too slow, less interaction, transmission ε=ε r ε 0 ε r =1 Hz cmx cmx c 14 8 10 14 10 105000 sec/103 ,sec,10 ≈==≈ − − ννλτ E-fields with frequencies greater than visible light frequency expected to be beyond influence of free electrons © E.A. Fitzgerald-1999 2 3.225 5 • Need Maxwell’s equations – from experiments: Gauss, Faraday, Ampere’s laws – second term in Ampere’s is from the unification – electromagnetic waves! Response of Light to Interaction with Material SI Units (MKS) MHB PED t D c J c Hx t B c Ex B D rrr rrr r rr r r r r π π π πρ 4 4 14 1 0 4 += += ∂ ∂ +=∇ ∂ ∂ −=∇ =•∇ =•∇ 00 00 0 ; 0 εεεµµµ µµµ εε ρ rr HMHB EPED t D JHx t B Ex B D == =+= =+= ∂ ∂ +=∇ ∂ ∂ −=∇ =•∇ =•∇ rrrr rrrr r rr r r r r Gaussian Units (CGS) © E.A. Fitzgerald-1999 3.225 6 Waves in Materials • Non-magnetic material, µ=µ 0 • Polarization non-existent or swamped by free electrons, P=0 t E JBx t B Ex ∂ ∂ +=∇ ∂ ∂ −=∇ r rr r r 000 εµµ t Bx Exx ∂ ∂∇ −=∇∇ r r )( 2 2 000 2 000 2 ][ t E t E E t E J t E ∂ ∂ + ∂ ∂ =∇ ∂ ∂ + ∂ ∂ −=∇− εµσµ εµµ For a typical wave, )()()( )( 2 000 2 0 )( 0 rErEirE erEeeEeEE titiriktrki ωεµσϖµ ϖϖϖ −−=∇ === −−•−• Wave Equation ωε σ ωε ωε ω 0 2 2 2 1)( )()()( i rE c rE += −=∇ )( )( )( 2 2 2 0 ωε ω ωε ω c k v c k eErE rik == = = • © E.A. Fitzgerald-1999 3 3.225 7 • Waves slow down in materials (depends on ε(ω)) • Wavelength decreases (depends on ε(ω)) • Frequency dependence in ε(ω) Waves in Materials )1( 11)( 0 0 0 ωτωε σ ωε σ ωε i ii − +=+= m ne i i p p 0 2 2 2 2 1)( ε ω τωω τω ωε = − += Plasma Frequency For ωτ >>>1, ε(ω) goes to 1 For an excellent conductor (σ 0 large), ignore 1, look at case for ωτ<<1 ω τω τωω τω ωε 2 2 2 )( pp i i i ≈ − ≈ © E.A. Fitzgerald-1999 3.225 8 Waves in Materials For a wave Let k=k real +k imaginary =k r +ik i )( 0 tkzi eEE ω − = [ zk tzki i r eeEE − − = ω 0 The skin depth can be defined by         +=       + = == 2 0 0 2 0 0 0 0 0 0 22 2 1 )( c i c i c k i cc k ε ωσ ε ωσ ωε σ ω ωε σ ω ωε ω δ ωµσσω ε δ ooo o i c k 2 2 1 2 === © E.A. Fitzgerald-1999 ] 4 3.225 9 Waves in Materials For a material with any σ 0 , look at case for ωτ>>1 () 2 2 1 ω ω ωε p −= ω<ω p , ε is negative, k=k i , wave reflected ω>ω p , ε is positive, k=k r , wave propagates R ω ω p © E.A. Fitzgerald-1999 3.225 10 Success and Failure of Free e- Picture • Success – Metal conductivity – Hall effect valence=1 – Skin Depth – Wiedmann-Franz law • Examples of Failure – Insulators, Semiconductors – Hall effect valence>1 – Thermoelectric effect – Colors of metals K/σ=thermal conduct./electrical conduct.~CT τ 2 3 1 thermv vc =Κ m Tk vnk T E c b thermb v v 3 ; 2 3 2 ==       ∂ ∂ = m Tnk m Tk nk bb b τ τ 2 2 3 3 2 3 3 1 =             =Κ m ne τ σ 2 = T e k b 2 2 3       = Κ σ Therefore : ~C! Luck: c vreal =c vclass /100; v real 2 =v class 2 *10 0 © E.A. Fitzgerald-1999 5 3.225 11 Wiedmann-Franz ‘Success’ Exposed Failure when c v and v 2 are not both in property Thermoelectric Effect TQE ∇= e nk ne nk ne c Q b b v 23 2 3 3 −= − =−= Thermopower Q is Thermopower is about 100 times too large! © E.A. Fitzgerald-1999 3.225 12 Waves in Vacuum 0, =ρ J 00 ; εεµµ == 2 2 00 2 t E E ∂ ∂ =∇ µ Wave Equation For typical wave: trik eEE ω−⋅ = 0 πνωλπ 2;2 == k 2 00 2 ωεµ= k ( 21 00 − == µνλω k For constant phase: ωt) →== ckv phase ω ⇒ • • ( 21 00 − = µc Example: Violet light (ν = 7.5 x 10 14 Hz) λ = c/ν = 400 nm k=2π/ λ = 1.57 x 10 7 m -1 ω = 2 πν = 4.71 x 10 15 s -1 After Livingston ε ) ε (kx- ) ε 6 3.225 13 Waves in Materials; Skin Depth δ The skin depth is defined by 21 2         = ωµσ δ ωµσµεω ik += 22 Conductive materials ( 21 ωµσ ik ≈ ( δ ωµσ ii + ±= + ±= 1 2 1 21 ( ( δδωω xxtitkxi eeEeEE −−−− ==⇒ 00 0 µµ≅ t E t E E ∂ ∂ + ∂ ∂ =∇ σµε 2 2 2 ; After Livingston ) ) ) ) µ 3.225 14 Plasma Frequency Remember: ωµσµεω ik += 22 where ( 1 1 00 >> − ≈ − = τ ωτ σ ωτ σ σ ii then         −=−≅ 2 2 2 0 22 1 ω ω µεω τ µσ µεω p k where ⇒       ≅ 21 2 ε ω m ne p Plasma Frequency For ω>ω p ; k is real number no attenuation! ω<ω p ; k contains imaginary component, wave reflected ⇒ Criteria for transparent electrode? ⇒ (Example: 8 x 10 27 /m 3 ; ω p = 4.3 x 10 15 s -1 ) After Livingston ) ω n=5. 7 . eEp = © E.A. Fitzgerald-1999 3 .22 5 2 Complex Representation of Waves sin(kx-ωt), cos(kx-ωt), and e -i(kx-ωt) are all waves e -i(kx-ωt) is the complex one and is the most general real. 2 0 0 2 0 0 0 0 0 0 22 2 1 )( c i c i c k i cc k ε ωσ ε ωσ ωε σ ω ωε σ ω ωε ω δ ωµσσω ε δ ooo o i c k 2 2 1 2 === © E.A. Fitzgerald-1999 ] 4 3 .22 5. ωε ω 0 2 2 2 1)( )()()( i rE c rE += −=∇ )( )( )( 2 2 2 0 ωε ω ωε ω c k v c k eErE rik == = = • © E.A. Fitzgerald-1999 3 3 .22 5 7 • Waves slow down in materials